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481

Documenta Math.

Rost Projectors and Steenrod Operations
Nikita Karpenko and Alexander Merkurjev1

Received: September 29, 2002
Communicated by Ulf Rehmann

Abstract. Let X be an anisotropic projective quadric possessing a
Rost projector ρ. We compute the 0-dimensional component of the
total Steenrod operation on the modulo 2 Chow group of the Rost
motive given by the projector ρ. The computation allows to determine
the whole Chow group of the Rost motive and the Chow group of every
excellent quadric (the results announced by Rost). On the other hand,
the computation is being applied to give a simpler proof of Vishik’s
theorem stating that the integer dim X + 1 is a power of 2.
2000 Mathematics Subject Classification: 11E04; 14C25
Keywords and Phrases: quadratic forms, Chow groups and motives,
Steenrod operations

M. Rost noticed that certain smooth projective anisotropic quadric hypersurfaces are decomposable in the category of Chow motives into a direct sum of
some motives. The (in some sense) smallest direct summands are called the
Rost motives. For example, the motive of a Pfister quadric is a direct sum
of Rost motives and their Tate twists. The Rost projectors split off the Rost
motives as direct summands of quadrics. In the present paper we study Rost
projectors by means of modulo 2 Steenrod operations on the Chow groups
of quadrics. The Steenrod operations in motivic cohomology were defined by
V. Voevodsky. We use results of P. Brosnan who found in [1] an elementary
construction of the Steenrod operations on the Chow groups.
As a consequence of our computations we give a description of the Chow groups
of a Rost motive (Corollary 8.2). This result (which has been announced by
M. Rost in [11]) allows to compute all the Chow groups of every excellent
quadric (see Remark 8.4).
We also give a simpler proof of a theorem of A. Vishik [3, th. 6.1] stating that
if an anisotropic quadric X possesses a Rost projector, then dim X + 1 is a
power of 2 (Theorem 5.1).
1

The second author was supported in part by NSF Grant #0098111.
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N. Karpenko and A. Merkurjev
Contents

1. Parity of binomial coefficients
2. Integral and modulo 2 Rost projectors
3. Steenrod operations
4. Main theorem
5. Dimensions of quadrics with Rost projectors
6. Rost motives
7. Motivic decompositions of excellent quadrics
8. Chow groups of Rost motives
References

482
482
484
485

488
488
490
491
493

1. Parity of binomial coefficients
1.1. Let i, n be any non-negative integers. The binomial coefficient
¡Lemma
¢
n+i
is
odd if and only if we don’t carry over units while adding n and i in
i
base 2.
Proof. For any integer a ≥ 0, let s2 (a) ¡be the
¢ sum of the digits in the base 2
is odd if and only if s2 (n + i) =
expansion of a. By [9, Lemma 5.4(a)], n+i
i

s2 (n) + s2 (i).
¤
The following statement is obvious:
Lemma 1.2. For any non-negative integer m, we don’t carry over units while
adding m and m + 1 in base 2 if and only if m + 1 is a power of 2.
¤
The following statement will be applied in the proof of Theorem 4.8:
Corollary
1.3. For any non-negative integer m, the binomial coefficient
¡−m−2¢
is
odd
if and only if m + 1 is a power of 2.
m
Proof. By Lemma 1.1, the binomial coefficient
µ
¶
µ
¶
−m − 2

m 2m + 1
= (−1)
m
m

is odd if and only if we don’t carry over units while adding m and m + 1 in
base 2. It remains to apply Lemma 1.2.
¤
2. Integral and modulo 2 Rost projectors
Let F be a field, X a quasi-projective smooth equidimensional variety over F .
We write CH(X) for the modulo 2 Chow group of X. The usual (integral)
Chow group is denoted by CH(X). We are working mostly with CH(X), but
several times we have to use CH(X) (for example, already the definition of a
modulo 2 Rost correspondence cannot be given on the level of the modulo 2
Chow group).
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Both groups are graded. We use the upper indices for the gradation by codimension of cycles and we use the lower indices for the gradation by the dimension
of cycles.
For projective X1 and X2 , an element ρ ∈ CH(X1 × X2 ) (we do not consider
the gradation on CH for the moment) can be viewed as a correspondence from
X1 to X2 ([2, §16.1]). In particular, it gives a homomorphism [2, def. 16.1.2]
¡
¢
ρ∗ : CH(X1 ) → CH(X2 ), ρ∗ (α) = pr 2∗ pr ∗1 (α) · ρ ,

where pr 1 and pr 2 are the two projections of X1 ×X2 onto X1 and X2 , and can
be composed with another correspondence ρ′ ∈ CH(X2 × X3 ) [2, def. 16.1.1].
The same can be said and defined with CH replaced by CH.
Starting from here, we always assume that char F 6= 2. Let ϕ be a nondegenerate quadratic form over F , and let X be the projective quadric ϕ = 0.
We set n = dim X = dim ϕ − 2 and we assume that n ≥ 1.
An element ̺ ∈ CHn (X × X) is called an (integral) Rost correspondence, if
over an algebraic closure F¯ of F one has:
¯ × x] + [x × X]
¯ ∈ CHn (X
¯ × X)

¯
̺F¯ = [X
¯ = XF¯ and a rational point x ∈ X.
¯ A Rost projector is a Rost correwith X
spondence which is an idempotent with respect to the composition of correspondences.
Remark 2.1. Assume that the quadric X is isotropic, i.e., contains a rational
closed point x ∈ X. Then [X × x] + [x × X] is a Rost projector. Moreover, this
is the unique Rost projector on X ([7, lemma 4.1]).
Remark 2.2. Let ̺ be a Rost correspondence on X. It follows from the Rost
nilpotence theorem ([12, prop. 1]) that a certain power of ̺ is a Rost projector
(see [7, cor. 3.2]). In particular, a quadric X possesses a Rost projector if and
only if it possesses a Rost correspondence.
A modulo 2 Rost correspondence ρ ∈ CHn (X×X) is a correspondence which can
be represented by an integral Rost correspondence. A modulo 2 Rost projector
is an idempotent modulo 2 Rost correspondence. Clearly, a modulo 2 Rost
correspondence represented by an integral Rost projector is a modulo 2 Rost
projector. Conversely,
Lemma 2.3. A modulo 2 Rost projector is represented by an integral Rost projector.
Proof. Let ρ be a modulo 2 Rost projector and let ̺ be an integral Rost correspondence representing ρ. The correspondence ̺F¯ is idempotent, therefore,
by the Rost nilpotence theorem (see [7, th. 3.1]), ̺r is idempotent for some

r; so, ̺r is an integral Rost projector. Since ρ is idempotent as well, ̺r still
represents ρ.
¤
Lemma 2.4. Let ̺ be an integral Rost correspondence and let ρ be a modulo
2 Rost correspondence. Then ̺∗ is the identity on CH0 (X) and on CH0 (X);
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N. Karpenko and A. Merkurjev

also ρ∗ is the identity on CH0 (X) and on CH0 (X). Moreover, for every i with
0 < i < n, the group ̺∗ CHi (X) vanishes over F¯ .
Proof. It suffices to prove the statements on ̺. Since CH0 (X) and CH0 (X)
inject into CH0 (XF¯ ) and CH0 (XF¯ ) (see [5, prop. 2.6] or [13] for the statement
on CH0 (X)), it suffices to consider the case where the quadric X has a rational
closed point x and ̺ = [X ×x]+[x×X]. Since [X ×x]∗ ([X]) = 0, [X ×x]∗ ([x]) =
[x], [x × X]∗ ([X]) = [X], [x × X]∗ ([x]) = 0 and since [X] generates CH0 (X)
while [x] generates CH0 (X), we are done with the statements on CH0 (X) and
on CH0 (X). Since [X × x]∗ ([Z]) = 0 = [x × X]∗ ([Z]) for any closed subvariety

Z ⊂ X of codimension 6= 0, n, we are done with the rest.
¤
3. Steenrod operations
In this section we briefly recall the basic properties of the Steenrod operations
on the modulo 2 Chow groups constructed in [1].
Let X be a smooth quasi-projective equidimensional variety over a field F .
For every i ≥ 0 there are certain homomorphisms S i : CH∗ (X) → CH∗+i (X)
called Steenrod operations; their sum (which is in fact finite because S i = 0 for
i > dim X)
S = SX = S 0 + S 1 + · · · : CH(X) → CH(X)
is the total Steenrod operation (we omit the ∗ in the notation of the Chow
group to indicate that S is not homogeneous). They have the following basic
properties (see [1] for the proofs): for any smooth quasi-projective F -scheme
X, the total operation S : CH(X) → CH(X) is a ring homomorphism such that
for every morphism f : Y → X of smooth quasi-projective F -schemes and for
every field extension E/F , the squares
S

SX

E
CH(Y ) −−−Y−→ CH(Y )
CH(XE ) −−−−
→ CH(XE )
x
x
x
x
f ∗

resE/F
and
resE/F 
f ∗




S

CH(X) −−−X−→ CH(X)

S

CH(X) −−−X−→ CH(X)

are commutative. Moreover, the restriction S i |CHn (X) is 0 for n < i and the
map α 7→ α2 for n = i; finally S 0 is the identity.
Also, the total Steenrod operation satisfies the following Riemann-Roch type
formula:
¡
¢
¡
¢
f∗ SY (α) · c(−TY ) = SX f∗ (α) · c(−TX )

(in other words, S modified by c(−T ) this way, commutes with the pushforwards) for any proper f : Y → X and any α ∈ CH(Y ), where f∗ : CH(Y ) →
CH(X) is the push-forward, c is the total Chern class, TX is the tangent bundle
of X, and c(−TX ) = c−1 (TX ) (the expression −TX makes sense if one considers
TX as an element of K0 (X)). This formula is proved in [1]. It also follows
from the previously formulated properties of S by the general Riemann-Roch
theorem of Panin [10].
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Lemma 3.1. Assume that X is projective. For any α ∈ CH(X) and for any
ρ ∈ CH(X × X), one has
¡
¢
SX (ρ∗ (α)) = SX×X (ρ)∗ SX (α) · c(−TX ) ,
where TX is the class in K0 (X) of the tangent bundle of X.

Proof. Let pr 1 , pr 2 : X × X → X be the first and the second projections. By
the Riemann-Roch formula applied to the morphism pr 2 , one has
³
´
¡
¢
SX (ρ∗ (α)) = pr 2 ∗ SX×X pr ∗1 (α) · ρ · c(−TX×X ) · c(TX ) .
By the projection formula for pr 2 , this gives
³
¡
¢´
SX (ρ∗ (α)) = pr 2 ∗ SX×X (pr ∗1 (α) · ρ) · c − TX×X + pr ∗2 (TX ) .

Since

TX×X = pr ∗1 (TX ) + pr ∗2 (TX ) ∈ K0 (X × X)
and since S (as well as c) commutes with the products and the pull-backs, we
get
³
´
´
³
¡
¢
pr 2 ∗ pr ∗1 SX (α) · c(−TX ) · SX×X (ρ) = SX×X (ρ)∗ SX (α) · c(−TX ) .

¤

4. Main theorem
In this section, let ϕ be an anisotropic quadratic form over F , and let X be the
projective quadric ϕ = 0 with n = dim X = dim ϕ − 2 ≥ 1. We are assuming
that an integral Rost projector (see §2 for the definition) ̺ ∈ CHn (X ×X) exists
for our X and we write ρ ∈ CHn (X × X) for the modulo 2 Rost projector. We
write h for the class in CH1 (X) (as well as in CH1 (X)) of a hyperplane section
of X.
Proposition 4.1. One has for every i ≥ 0:
¡
¢
¡
¢
S ρ∗ (hi ) = S(ρ)∗ hi · (1 + h)i−n−2 .

Proof. Since h ∈ CH1 (X), we have S(h) = S 0 (h) + S 1 (h). Since S 0 = id while
S 1 on CH1 (X) is the squaring, S(h) = h+h2 = h(1+h), hence S(hi ) = S(h)i =
hi (1 + h)i . To finish the proof it suffices to check that c(TX ) = (1 + h)n+2 for
the tangent bundle TX of the quadric X: then the formula of Lemma 3.1 will
give the formula of Proposition 4.1.
Let i : X ֒→ P be the embedding of X into the (n + 1)-dimensional projective
space P . Let us write H for the class in CH1 (P ) of a hyperplane. Note that
h = i∗ (H).
The exact sequence of vector X-bundles
0 → TX → i∗ (TP ) → i∗ (OP (2)) → 0
¡
¢
¡
¢
gives the equality c(TX )·i∗ c(OP (2)) = i∗ c(TP ) . Since c(OP (2)) = 1+2H =
1 (we are working with the modulo 2 Chow groups) and c(TP ) = (1 + H)n+2 ,
we get c(TX ) = (1 + h)n+2 .
¤
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Lemma 4.2. Let L/F be a field extension such that the quadric XL is isotropic.
Then S i (ρL ) = 0 for every i > 0.
Proof. By the uniqueness of a modulo 2 Rost projector on an isotropic quadric
(Remark 2.1) ρL = [X] × [x] + [x] × [X], where x ∈ XL is a rational point.
Since S = S 0 = id on CH0 (XL ) ∋ [X] as well as on CHn (XL ) ∋ [x], we have
S(ρL ) = S([X] × [x] + [x] × [X]) = S([X]) × S([x])+
S([x]) × S([X]) = [X] × [x] + [x] × [X] = ρL = S 0 (ρL ) .
¤
Lemma 4.3. The Witt index of the quadratic form ϕF (X) is 1.
Proof. Let Y ⊂ X be a subquadric of codimension 1. If iW (ϕF (X) ) > 1, Y
has a rational point over F (X). Therefore there exists a rational morphism
X → Y . Let α ∈ CHn (X × X) be the correspondence given by the closure
of the graph of this morphism. Let us show that ∆∗ (̺ ◦ α) ∈ CH0 (X), where
∆ : X → X × X is the diagonal morphism, is an element of CH0 (X) of degree
1 (giving a contradiction with the fact that the quadric X is anisotropic, this
will finish the proof).
Clearly, verifying the assertion on the degree, we may replace F by a field
extension of F . Therefore, we may assume that X has a rational point x. In
this case ̺ = [X × x] + [x × X]. Since [x × X] ◦ α = 0 (because dim Y < dim X)
while [X × x] ◦ α = [X × x], we get ∆∗ (̺ ◦ α) = [x].
¤
Lemma 4.4. Let X be an anisotropic F -quadric such that the Witt index of the
quadratic form ϕF (X) is 1. Then for every α ∈ CHi (XF (X) ), i > 0, the degree
of the 0-cycle class hi · α is even.
Proof. It is sufficient to consider the case i = 1. We have ϕF (X) ≃ ψ⊥H for an
anisotropic quadratic form ψ over F (X) (where H is a hyperbolic plane). Let
X ′ be the quadric ψ = 0 over F (X). There is an isomorphism [5, §2.2]
f : CH1 (XF (X) ) → CH0 (X ′ )
taking hn−1 to the class of a closed point of degree 2. Since the quadric X ′ is
anisotropic, the group CH0 (X ′ ) is generated by the class of a degree 2 closed
point (see [5, prop. 2.6] or [13]); therefore the group CH1 (XF (X) ) is generated
by hn−1 . Since deg(h · hn−1 ) = 2, it follows that the integer deg(h · α) is even
for every α ∈ CH1 (XF (X) ).
¤
For the modulo 2 Chow groups we get
Corollary 4.5. Let µ ∈ CHi (X × X) for some 0 < i ≤ n be a correspondence
such that µF (X) = 0. Then µ∗ (hi ) = 0.
Proof. We replace µ by its representative in CHi (X ×X) and we mean by h the
integral class of a hyperplane section of X in the proof (while in the statement
h is the class of a hyperplane section in the modulo 2 Chow group). Since the
degree homomorphism deg : CHn (X) → Z is injective ([5, prop. 2.6] or [13])
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with the image 2Z, it suffices to show that deg(µ∗ (hi )) is divisible¡by 4. Let us
¢
compute this degree. By definition of µ∗ , we have µ∗ (hi ) = pr 2 ∗ µ · pr ∗1 (hi ) .
Note that the product µ · pr ∗1 (hi ) is in CH0 (X × X) and the square
pr 2∗

CH0 (X × X) −−−−→ CH0 (X)



pr 1∗ 
degy
y
CH0 (X)

deg

−−−−→

Z

commutes (the two compositions being the degree homomorphism of the group
CH0 ¡(X × X)). ¢Therefore the degree of µ∗ (hi ) coincides with the degree of
pr 1 ∗ µ · pr ∗1 (hi ) . By the projection formula for pr 1 ∗ the latter element coincides with the product hi · pr 1∗ (µ).
We are going to check that the degree of this element is divisible by 4. Since the
degree does not change under extensions of the base field, it suffices to verify
the divisibility relation over F (X). The class pr 1∗ (µ)F (X) is divisible by 2 by
assumption, therefore the statement follows from Lemmas 4.3 and 4.4.
¤
Corollary 4.6. S n−i (ρ)∗ (hi ) = 0 for every i with 0 < i < n.
Proof. We take µ = S n−i (ρ). Since i < n, we have µF (X) = 0 by Lemma 4.2.
Since i > 0, we may apply Corollary 4.5 obtaining µ∗ (hi ) = 0.
¤
Putting together Corollary 4.6 and Proposition 4.1, we get
Corollary 4.7. For every i > 0, one has:
µ
¶
¡
¢
i−n−2
S n−i ρ∗ (hi ) =
· ρ∗ (hn ) .
n−i
¡
¢
Proof.¡ By Proposition 4.1,
S n−i ρ∗ (hi ) is the n-codimensional component of
¢
S(ρ)∗ hi · (1 + h)i−n−2 ; moreover, according to Corollary 4.6, S(ρ) can be
replaced by S 0 (ρ) = ρ.
¤
Finally, by Corollary 1.3, computing the binomial coefficient modulo 2, together
with Lemma 2.4, computing ρ∗ (hn ), we get
Theorem 4.8. Suppose that the anisotropic quadric X of dimension n possesses a Rost projector. Let ρ be a modulo 2 Rost projector on X and let i be
an integer with 0 < i < n. Then
¡
¢
S n−i ρ∗ (hi ) = hn
in CH0 (X) if (and only if ) the integer n − i + 1 is a power of 2.

¤

As the quadric X is anisotropic, CH0 (X) is an infinite cyclic group generated
by hn (see [5, prop. 2.6] or [13]); in particular, hn in CH0 (X) is not 0. Therefore
we get
Corollary 4.9. For every i such that 0 < i < n and n − i + 1 is a power of
2, the element S n−i (ρ∗ (hi )) (and consequently ρ∗ (hi )) is non-zero.
¤
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5. Dimensions of quadrics with Rost projectors

The following Theorem is proved in [3]. The proof given there makes use of
the Steenrod operations in the motivic cohomology constructed by Voevodsky
(since Voevodsky has announced that the operations were constructed in any
characteristic 6= 2 only quite recently, the assumption char F = 0 was made in
[3]). Here we give an elementary proof.
Theorem 5.1 ([3, th. 6.1]). If X is an anisotropic smooth projective quadric
possessing a Rost projector, then dim X + 1 is a power of 2.
Proof. Let us assume that this is not the case. Let r be the largest integer such
r
that n > 2r −1 where
¡ n i=¢dim X. Then Theorem 4.8 applies to i = n−(2 −1),
n−i
stating that S
ρ∗ (h ) 6= 0. Note that n − i ≥ i. Since the Steenrod
operation S i is¡ trivial¢ on CHj (X) with i > j, it follows that n − i = i and
therefore S n−i ρ∗ (hi ) = ρ∗ (hi )2 . Since the element ̺∗ (hi ) (where ̺ is the
integral Rost projector) vanishes over F¯ (Lemma 2.4), its square vanishes over
i 2
F¯ as well. The group
¡ CHi0 (X)
¢ injects however into CH0 (XF¯ ), hence ̺∗ (h ) = 0
n−i
and therefore S
ρ∗ (h ) = 0, giving a contradiction with Corollary 4.9. ¤

Remark 5.2. It turns out that Theorem 5.1 is extremely useful in the theory of
quadratic forms. For example, it is the main ingredient of Vishik’s proof of the
theorem that there is no anisotropic quadratic forms satisfying 2r < dim ϕ <
2r + 2r−1 and [ϕ] ∈ I r (F ) (see [14], [15]).
6. Rost motives
Let Λ be an associative commutative ring with 1. We set ΛCH = Λ ⊗Z CH (we
will only need Λ = Z or Λ = Z/2).
We briefly recall the construction of the category of Grothendieck ΛCH-motives
similar to that of [4]. A motive is a triple (X, p, n), where X is a smooth
projective equidimensional F -variety, p ∈ ΛCHdim X (X × X) an idempotent
correspondence, and n an integer. Sometimes the reduced notations are used:
(X, n) for (X, p, n) with p the diagonal class; (X, p) for (X, p, n) with n = 0;
and (X) for (X, 0), the motive of the variety X.
For a motive M = (X, p, n) and an integer m, the m-th twist M (m) of M is
defined as (X, p, n + m).
The set of morphisms is defined as
¡
¢
′
′
Hom (X, p, n), (X ′ , p′ , n′ ) = p′ ◦ ΛCHdim X −n+n (X × X ′ ) ◦ p .

In particular, every homogeneous correspondence α ∈ ΛCH(X ×X ′ ) determines
a morphism of every twist of (X, p) to a certain twist of (X ′ , p′ ).
The Chow group ΛCH∗ (X, p, n) of a motive (X, p, n) is defined as
ΛCH∗ (X, p, n) = p∗ ΛCH∗−n (X) .
It gives an additive functor of the category of ΛCH-motives to the category
of graded abelian groups (namely, the functor Hom(M (∗), −), where M is the
motive of a point).
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For any Λ, there is an evident additive functor of the category of CH-motives to
the category of ΛCH-motives (identical on the motives of varieties). In particular, every isomorphism of CH-motives automatically produces an isomorphism
of the corresponding ΛCH-motives. This is why bellow we mostly formulate
the results only on the integral motives.
We are coming back to the quadratic forms.
Definition 6.1. Let ̺ be an integral Rost projector on a projective quadric
X. We refer to the motive (X, ̺) as to an (integral) Rost motive. (While the
CH-motive given by a modulo 2 Rost projector can be called a modulo 2 Rost
motive.) A Rost motive is anisotropic, if the quadric X is so.
Let now π be a Pfister form and let ϕ be a neighbor of π which is minimal, that
is, has dimension dim π/2 + 1. As noticed by M. Rost (see [7, 5.2] for a proof),
the projective quadric X given by ϕ possesses an integral Rost projector ̺.
Proposition 6.2. Let ϕ be as above. Let ̺′ be the Rost projector on the
quadric X ′ given by a minimal neighbor ϕ′ of another Pfister form π ′ . The
Rost motives (X, ̺) and (X ′ , ̺′ ) are isomorphic if and only if the Pfister forms
π and π ′ are isomorphic.
Proof. First we assume that (X, ̺) ≃ (X ′ , ̺′ ). Looking at the degrees of 0cycles on X and on X ′ , we see that ϕ is isotropic if and only if ϕ′ is isotropic,
thus π is isotropic if and only if π ′ is isotropic. Therefore, the forms πF (π′ ) and
πF′ (π) are isotropic. Since π and π ′ are Pfister forms, it follows that π ≃ π ′ .
Conversely, assume that π ≃ π ′ . By [12, §3], in order to show that (X, ̺) ≃
(X ′ , ̺′ ), it suffices to construct a morphism of motives (X, ̺) → (X ′ , ̺′ ) which
becomes mutually inverse isomorphism over an algebraic closure F¯ of F . We
will do a little bit more: we construct two morphisms (X, ̺) ⇄ (X ′ , ̺′ ) which
become mutually inverse isomorphisms over an algebraic closure F¯ of F (in
this case, the initial F -morphisms are isomorphisms, although possibly not
mutually inverse ones, [7, cor. 3.3]).
Since π ≃ π ′ , the quadratic forms ϕ′F (ϕ) and ϕF (ϕ′ ) are isotropic. Therefore
there exist rational morphisms X → X ′ and X ′ → X. The closures of their
graphs give two correspondences α ∈ CH(X × X ′ ) and β ∈ CH(X ′ × X).
Over F¯ we have: ̺′ ◦ α ◦ ̺ = [X × x′ ] + a[x × X ′ ], where x ∈ XF¯ and x′ ∈ XF′¯
are closed rational points, while a is an integer (which coincides, in fact, with
the degree of the rational morphism X → X ′ ). Similarly, ̺ ◦ β ◦ ̺′ = [X ′ × x] +
b[x′ × X] with some b ∈ Z over F¯ .
We are going to check that the integers a and b are odd. For this we consider
the composition
(̺ ◦ β ◦ ̺′ ) ◦ (̺′ ◦ α ◦ ̺) ∈ CH(X × X) .
Over F¯ this composition gives [X × x] + ab[x × X]. Consequently, by [6, th. 6.4]
and Lemma 4.3, the integer ab is odd.
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Let us take now
a−1
b−1
· [y × X ′ ] and β ′ = β −
· [y ′ × X]
2
2
with some degree 2 closed points y ∈ X and y ′ ∈ X ′ . Then over F¯
α′ = α −

̺′ ◦ α′ ◦ ̺ = [X × x′ ] + [x × X ′ ] while ̺ ◦ β ′ ◦ ̺′ = [X ′ × x] + [x′ × X] ,
therefore the two F -morphisms (X, ̺) ⇄ (X ′ , ̺′ ) given by these α′ and β ′
become mutually inverse isomorphisms over F¯ .
¤
Definition 6.3. The motive (X, ̺) for X and ̺ as in Proposition 6.2 (more
precisely, the isomorphism class of motives) is called the Rost motive of the
Pfister form π and denoted R(π).
Remark 6.4. It is conjectured in [7, conj. 1.6] (with a proof given for 3 and
7-dimensional quadrics) that every anisotropic Rost motive is the Rost motive
of some Pfister form.
7. Motivic decompositions of excellent quadrics
Theorem 7.1 (announced in [11]). Let ϕ be a neighbor of a Pfister form π
and let ϕ′ be the complementary form (that is, ϕ′ is such that the form ϕ⊥ϕ′
is similar to π). Then
´L
³ m−1
L
R(π)(i)
(X) ≃
(X ′ )(m) ,
i=0

′

where m = (dim ϕ − dim ϕ )/2, X is the quadric defined by ϕ, and X ′ is the
quadric defined by ϕ′ .
Proof. Similar to [12, th. 17] (see also [7, prop. 5.3]).

¤

We recall that a quadratic form ϕ over F is called excellent, if for every field
extension E/F the anisotropic part of the form ϕE is defined over F . An
anisotropic quadratic form is excellent if and only if it is a Pfister neighbor
whose complementary form is excellent as well [8, §7].
Let π0 ⊃ π1 ⊃ · · · ⊃ πr be a strictly decreasing sequence of embedded Pfister
forms. Let ϕ be the quadratic form such that the class [ϕ] of ϕ in the Witt ring
of F is the alternating sum [π0 ]−[π1 ]+· · ·+(−1)r [πr ], while the dimension of ϕ
is the alternating sum of the dimensions of the Pfister forms. Clearly, ϕ is excellent. Moreover, every anisotropic excellent quadratic form is similar to a form
obtained this way. Let us require additionally that 2 dim πr < dim πr−1 . Then
every anisotropic excellent quadratic form is still similar to a form obtained
this way and, moreover, the Pfister forms π0 , . . . , πr are uniquely determined
by the initial excellent quadratic form.
Let X be an excellent quadric, that is, the quadratic form ϕ giving X is excellent. As Theorem 7.1 shows, the motive of X is a direct sum of twisted Rost
motives. More precisely,
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Corollary 7.2 (announced in [11]). Let X be the excellent quadric determined
by Pfister forms π0 ⊃ · · · ⊃ πr . Then
(X) ≃

³ mL
0 −1
i=0

R(π0 )(i)

´ L ³ m0 +m
L1 −1

R(π1 )(i)

i=m0

...

´L

L³

...
m0 +···+m
L r

i=m0 +···+mr−1

R(πr )(i)

´

with mj = dim πj /2 − dim πj+1 + dim πj+2 − . . . .

¤

Here are three examples of excellent forms which are most important for us:
Example 7.3 (Pfister forms, [12, prop. 19]). Let ϕ = π be a Pfister form.
Then
dimL
π/2−1
(X) ≃
R(π)(i) .
i=0

Example 7.4 (Maximal neighbors, [12, th. 17]). Let ϕ be a maximal neighbor of a Pfister form π (that is, dim ϕ = dim π − 1). Then
(X) ≃

dimL
π/2−2

R(π)(i)

i=0

Example 7.5 (Norm forms, [12, th. 17]). Let ϕ be a norm quadratic form,
that is, ϕ is a minimal neighbor of a Pfister form π containing a 1-codimensional
subform which is similar to a Pfister form π ′ . Then
′
´
L ³ dim πL/2−1
R(π ′ )(i) .
(X) ≃ R(π)
i=1

8. Chow groups of Rost motives
The following theorem computes the Chow groups of the modulo 2 Rost motive
of a Pfister form.
Theorem 8.1 (announced in [11]). Let ρ be the modulo 2 Rost projector on
the projective n-dimensional quadric X given by an anisotropic minimal Pfister
neighbor. Let i be an integer with 0 ≤ i ≤ n. If i + 1 is a power of 2, then the
Chow group CHi (X, ρ) = ρ∗ CHi (X) is cyclic of order 2 generated by ρ∗ (hn−i ).
Otherwise this group is 0.
Proof. According to Proposition 6.2, we may assume that X is a norm quadric,
that is, X contains a 1-codimensional subquadric Y being a Pfister quadric.
Let r be the integer such that n = dim X = 2r − 1.
We proceed by induction on r. Let Y ′ ⊂ Y be a subquadric of dimension
2r−1 − 2 which is a Pfister quadric. Let X ′ be a norm quadric of dimension
2r−1 − 1 such that Y ′ ⊂ X ′ ⊂ Y . Let ρ′ be a modulo 2 Rost projector on X ′ .
By Example 7.5, passing from CH-motives to the category of CH-motives, we
see that the motive of X is the direct sum of the motive (X, ρ) and the motives
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N. Karpenko and A. Merkurjev

(X ′ , ρ′ , i) with i = 1, . . . , 2r−1 − 1. Therefore
r−1
´
L ³ 2 L−1 ′
ρ∗ CH(X ′ )
CH(X) ≃ ρ∗ CH(X)
i=1

(we do not care about the gradations on the Chow groups).
Also the motive of Y decomposes in the direct sum of the motives (X ′ , ρ′ , i)
with i = 0, . . . , 2r−1 − 1 (Example 7.3). Therefore
CH(Y ) ≃

2r−1
L−1

ρ′∗ CH(X ′ ) .

i=0

It follows that the order of the group CH(Y ) is |ρ′∗ CH(X ′ )|2
r−1
of CH(X) is |ρ′∗ CH(X ′ )|2 −1 · |ρ∗ CH(X)|.
In the exact sequence

r−1

, while the order

CH(Y ) → CH(X) → CH(U ) → 0
with U = X \ Y , the Chow group CH(U ) of the affine norm quadric U is
computed by M. Rost ([7, th. A.4]): CH(U ) = CH0 (U ) ≃ Z/2. Therefore, the
orders of these groups satisfy
| CH(X)| ≤ | CH(Y )| · | CH(U )| = 2| CH(Y )| ,
thus |ρ∗ CH(X)| ≤ 2|ρ′∗ CH(X ′ )|.
The group ρ′∗ CH(X ′ ) is known by induction. In particular, the order of this
group is 2r . It follows that the order of ρ∗ CH(X) is at most 2r+1 . Corollary 4.9
gives already r+1 non-zero elements of ρ∗ CH∗ (X) living in different dimensions
s
(more precisely, ρ∗ (hn−2 +1 ) 6= 0 for s = 1, . . . , r − 1 by Corollary 4.9 and for
s = 0, r by Lemma 2.4) and therefore generating a subgroup of order 2r+1 . It
follows that the order of ρ∗ CH(X) is precisely 2r+1 and the non-zero elements
we have found generate the group ρ∗ CH(X).
¤
The integral version of Theorem 8.1 is given by
Corollary 8.2 (announced in [11]). For X as in Theorem 8.1, let ̺ be the
integral Rost projector on X. Then for every i with 0 ≤ i ≤ n, the Chow group
CHi (X, ̺) = ̺∗ CHi (X) is a cyclic group generated by ̺∗ (hn−i ). Moreover, the
element ̺∗ (hn−i ) is
• 0, if i + 1 is not a power of 2;
• of order 2, if i + 1 is a power of 2 and i 6∈ {0, n};
• of infinite order, if i ∈ {0, n}.
Proof. The statements on CHn (X) and on CH0 (X) are clear. The rest follows from Theorem 8.1, if we show that 2 · ̺∗ CHi (X) = 0 for every i with
0 < i < n. Let L/F be a quadratic extension such that XL is isotropic.
Then (̺L )∗ CHi (XL ) = 0 for such i by [7, cor. 4.2] (cf. Lemma 2.4). Since
the composition of the restriction CHi (X) → CHi (XL ) with the transfer
CHi (XL ) → CHi (X) coincides with the multiplication by 2, it follows that
2 · ̺∗ CHi (X) = 0.
¤
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Remark 8.3. The result of Corollary 8.2 was announced in [11]. A proof has
never appeared.
Remark 8.4. Clearly, Corollary 8.2 describes the Chow group of the Rost
motive of an anisotropic Pfister form. Since the motive of any anisotropic
excellent quadric is a direct sum of twists of such Rost motives (Corollary
7.2), we have computed the Chow group of an arbitrary anisotropic excellent
projective quadric. Note that the answer depends only on the dimension of the
quadric.
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Nikita Karpenko
Laboratoire des Math´ematiques
Facult´e des Sciences
Universit´e d’Artois
rue Jean Souvraz SP 18
62307 Lens Cedex
France
karpenko@euler.univ-artois.fr

Alexander Merkurjev
Department of Mathematics
University of California
Los Angeles, CA 90095-1555
USA
merkurev@math.ucla.edu

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